Deform a flat shape while retaining the original mesh
Yes, this is possible:

For manual control:


Rigging (use bones to bend and fold the shape)
Shape key (manually make some edits - while keeping the base shape) - the shapes can be mixed together and their influences animated.
Check on other deform modifiers too, shrink-wrap. lattice... all of them really, could come in handy for deforming a flat shape.


For effects:


Wave modifier
Warp modifier
Displace (consider using texture to displace, it can be animated too).
Physics: (also worth looking into)


Soft body
Cloth





Note, vertex groups can be used to influence many of the distortion methods described above.



This question is quite general (thats OK), but means my answer is generalized too.